Covariants

Date: 05/02/97 at 11:09:14
From: Vincent
Subject: Trigonometry
Dr. Math,
Please help me answer the following trigonometry problem:
Express tan(pi/2 + X) as a single function of X or as a constant.

Date: 05/02/97 at 11:45:34
From: Doctor Wilkinson
Subject: Re: Trigonometry
This problem involves the relations between the various trig functions
and the "co" variants. For example, sine and cosine, tangent and
cotangent, secant and cosecant. The "co" here relates to
complementary angles. The sine of an angle is the cosine of the
complement of the angle, and so on. So we have the formulas:
sin(pi/2 - x) = cos(x)
tan(pi/2 - x) = cot(x)
sec(pi/2 - x) = csc(x)
In your example, you have:
tan(pi/2 + X)
We can apply the second formula with x = -X to get:
tan(pi/2 + X) = cot(-X)
Now we are close. The cot(-X) = -cot(X). Remember the cotangent is
the quotient of the sine and the cosine, and we have the formulas:
sin(-x) = -sin(x)
cos(-x) = cos(x)
So cot(-x) = cos(x)/(-sin(x)) = -cos(x)/sin(x) = -cot(x)
-Doctor Wilkinson, The Math Forum
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